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I have a question concerning the proof of the following theorem:

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I don't see how they can state that we're looking for a unique solution $M$; $$ f(x)=\sum_{k=0}^{n-1}\frac{f^{(k)}(c)}{k!}(x-c)^k+\frac{M(x-c)^n}{n!}. $$ How do we know the solution is unique? How do we know it exists at all? I'm a little bit confused by this. Why don't they word it like this: we are looking for a solution $M$, and we're hoping that it exists and is unique.

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    The proof is confusing, yes. Take a look to [these other proofs](https://proofwiki.org/wiki/Taylor%27s_Theorem/One_Variable), or just search for a proof from a different book of analysis. [These](http://math.stackexchange.com/questions/481661/simplest-proof-of-taylors-theorem#492165) other proofs are useful too. – 2017-02-06

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